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Santino Bautista

Santino Bautista

Answered question

2022-06-09

tan A 2 + tan B 2 + tan C 2 4 3

Answer & Explanation

Zayden Andrade

Zayden Andrade

Beginner2022-06-10Added 22 answers

Let tan A 2 = x , tan B 2 = y , tan C 2 = z , tan A 2 + tan B 2 + tan C 2 = w. WLOG x 3
Note that we have the identity tan ( A + B 2 ) = tan ( π C 2 )
x + y 1 x y = 1 z
x y + y z + z x = 1 ( y + z ) 2 4 + x ( w x ) = ( w x ) 2 4 + x ( w x )
This gives us
w 4 + 4 x 2 x 4 3
From the fact that w 4 + 4 x 2 x 4 3 is an increasing function.

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